Output-based disturbance rejection control for 1-D anti-stable Schrödinger equation with boundary input matched unknown disturbance

In this paper, we are concerned with the output feedback control design for a system (plant) described by a boundary controlled anti-stable one-dimensional Schrödinger equation. Our output measure signals are the displacements at both side. An untraditional infinite-dimensional disturbance estimator is developed to estimate the disturbance. Based on the estimator, we propose a state observer that is exponentially convergent to the original system and then design a stabilizing control law consisting of two parts: The first part is to compensate the disturbance by using its approximated value and the second part is to stabilize the observer system by applying the classical backstepping approach. The resulting closed-loop system is shown to be exponentially stable with guaranteeing that all internal systems are uniformly bounded. An effective output-based disturbance rejection control algorithm is concluded. An application, namely, a cascade of ODE–wave systems, is investigated by the developed control algorithm. Numerical experiments are carried out to illustrate the effectiveness of the proposed control law.